Electrons in Solids
Energy Bands and Resistance in
Conductors and Semiconductors
What Have We Learned About
Electrical Storage
• The electric force FE on a charge q0 can be considered due
to an electric field which is produced by other charges in
the area
FE = q0 E
• If moving a charge between two points requires work (or
does work), the charge gains (or loses) potential energy:
DV = –  E  dx = (for a constant field) EDx
• Capacitors store charge Q in proportion to the voltage V
between the plates:
C = Q/V = C = e0 A/d
• Capacitors are used in RAM
What Have We Learned About
Magnetic Storage?
• Two domains magnetized in same direction is a 0
• Two domains magnetized in opposite directions is
a1
• Direction of magnetization changes at start of new
bit.
• Magnetic data is written by running a current
through a loop of wire near the disk
• As magnetic data passes by coil of wire, changing
field induces currents according to Faraday’s Law:
e
d B
dB
 iR  
 A
dt
dt
What Have We Learned About
Magnetoresistance?
• Charges traveling through magnetic field experience
magnetic force (provided velocity and field are not
aligned):
FB = qv x B = (if v perpendicular to B) qvB
• In a current-carrying wire, this force results in more
frequent collisions and thus an increased resistance:
Magnetoresistance
• Electrons traveling through magnetized material undergo
spin-dependent scattering
• When magnetic field is present in magnetic superlattice,
scattering of electrons is cut dramatically, greatly
decreasing resistance: Giant magnetoresistanced
Stuff to remember about GMR
• Electrons (and other elementary “particles”) have
intrinsic magnetic fields, identified by spin
• The scattering of electrons in a ferromagnetic
material depends on the spin of the electrons
• Layers of ferromagnetic material with alternating
directions of magnetization exhibit maximum
resistance
• In presence of magnetic field, all layers align and
resistance is minimized
What Have We Learned About
Spectra?
• ENERGY LEVELS ARE QUANTIZED
• Different elements have different allowed energies (since
different numbers of protons and electrons provide
different structure of attraction
• Light emitted when electrons move from a high energy
level to a lower energy level in an atom will have only
certain, QUANTIZED, allowed energies and wavelengths.
• Those wavelengths depend solely on the element emitting
the light and compose the characteristic emission spectrum
for that element
Our Model of the Atom
• If the atom is in the “ground state” of lowest energy, electrons fill the
states in the lowest available energy levels. The first shell has two
possible states, and the second shell has eight possible states. Higher
shells have more states, but we’ll represent them with the eight states
in the first two sub-shells.
• Electrons in the outermost shell are called “valence” electrons. We’ll
make them green to distinguish from e- in filled shells
E=0 (unbound)
n=4
n=3
n=2
n=1
Really eight distinct states with
closely spaced energies, since two
electrons cannot occupy the same
state.
The Hydrogen Atom
• Has one electron, normally in the ground state n=1
• This electron can absorb energy and go to a higher state, like n=3
• The atom will eventually return to its ground state, and the electron
will emit the extra energy in the form of light.
• This light will have energy E = (13.6 ev)(1/1 – 1/32) = 12.1 eV
• The corresponding wavelength is l = hc/E = 1020 Å
E=0 (unbound)
n=4
n=3
n=2
n=1
Other Atoms
• Electrons can absorb energy and move to a higher level
– White light (all colors combined) passing through a gas will come out
missing certain wavelengths (absorption spectrum)
• Electrons can emit light and move to a lower level
• Calculating the allowed energies extremely complicated for anything
with more than one electron
• But can deduce allowed energies from light that is emitted
n=4
n=3
n=2
n=1
E=0 (unbound)
Really eight closely spaced
energies, since no two electrons
can occupy same state
Atomic Bonding
• Electrons in an unfilled valence shell are loosely bound
• Atoms will form bonds to fill valence shells, either by
sharing valence electrons, borrowing them, or loaning
them
• When atoms bond in solids, sharing electrons, each
atom’s energy levels get slightly shifted
E=0 (unbound)
n=4
n=3
n=2
n=1
Electron Motion
• Electrons can only move to an open energy state
• If the atom does not absorb energy, electrons can
only move to an open energy state in the same
shell
(drawing is NOT to scale)
E=0 (unbound)
n=4
n=3
n=2
n=1
Electrons in Solids
• The shifted energies in adjacent atoms combine to create a continuous
“band” of allowed energies for each original energy level; each band,
however, has a finite number of states equal to the number in original
atoms
• Electrons can move from the locality of one atom to the next only if an
energy state is available within the same band
Conductors & Semiconductors
• In conductors, the valence band is only partiallyfull, so electrons can easily move from being near
one atom to being near another
• In semiconductors and insulators, the valence
band is completely full, so electrons must gain
extra energy to move
• In semiconductors, extra electrons (or holes) can
be introduced in a “controlled” way.
Electrons in an Electric Field
• Conduction electrons move randomly in all directions in
the absence of a field.
• If a field is applied, the electric force results in acceleration
in a particular direction:
F=ma= –eE
 a = –eE/m
• As the charges accelerate, the potential energy stored in the
electric field is converted to kinetic energy which can be
converted into heat and light as the electrons collide with
atoms in the wire
• This acceleration produces a velocity
v = at = –eEt/m
Electrical current
• If an electric field points from left to right,
positive charge carriers will move toward the
right
while negative charges will move toward the
left
• The result of both is a net flow of positive charge
to the right.
• Current is the net change in positive charge per
time
DQ
i
Dt
Electrical current
• Look at the movement of charges through a wire
with a potential difference applied - animation
• The net velocity is called drift velocity vd
• The charge contained in a cross-sectional volume
is Dq=Nq(Volume)=NqADx=NqAvdDt
• So the charge per time crossing through A is
DQ NqAvd Dt
i

 NqAvd
Dt
Dt
How do I figure the drift velocity?
• The drift velocity is the net velocity of
charge carriers after collisions
• It is not equal to the velocity caused by
acceleration of field on individual charge,
but it certainly is proportional to it.
• Since current is proportional to vd, current is
proportional to electric field
qE
v
t  vd
m
Answer the
Ohm’s Law - Before You Start
Questions in Today’s Activity,
Then Continue Through
Question 4
Ohm’s Law
•
•
Electric field is proportional to potential difference or
voltage
So, . . .
i  vd  v  E  V  i  V
Ohm gets credit for
being the first to
notice that many
materials displayed
this proportionality. He
defined resistance as
the ratio of V to i.
Ohm’s Law - worth remembering
V = iR
• Ohm’s law only applies to materials in which the
behavior of charge carriers can be described
statistically by the drift velocity
• In short, Ohm’s law only applies to ohmic
materials.
• Were both of your resistors made of ohmic
material?
A Good Analogy to Remember
On what does resistance depend?
On What Does Resistance Depend?
• If I increase the length of a wire, the current flow
decreases because of the longer path
• If I increase the area of a wire, the current flow
increases because of the wider path
R = r L/A
• If I change to a material with better conductivity,
the current flow
increases because charge carriers move better
• If I change the temperature, the current flow
changes
Conductors vs. Semiconductors
• Electrons are free to move in the conduction band
• As temperatures rise, electrons collide more with vibrating
atoms; this effect reduces current
• Conductors have a partially-filled valence band which
doubles as a conduction band at any temperature
– The primary effect of temperature on resistance is due to more
collisions at higher temperatures
• Semiconductors have a completely-filled valence band, so
electrons have to be “excited” to enter the conduction band
– The primary effect of temperature on resistance is due to this
requirement: the higher the temperature, the more conduction
electrons
What is a semiconductor?
• Metals
– Many free electrons not tied up in chemical
bonds
• Insulators
– All electrons (in intrinsic material) tied up in
chemical bonds
Crystal (Perfect)
Crystal (Excited)
Crystal (Excited)
Band Gap
Energy
Conduction Band
Band Gap Energy Eg
(Minimum Energy needed to
break the chemical bonds)
Valence Band
Position
Band Gap
Energy
Conduction Band
h  E g
photon
in
Valence Band
Position
Band Gap
Energy
Conduction Band
photon out
Valence Band
Position
Band Gap
Energy
Conduction Band
photon out
Valence Band
Position
What have we learned today?
• When atoms bind, the energy levels in each atom get shifted
slightly (the size of the shift is very small compared with the
energy difference between different levels)
• Atoms in solids form bands of closely-spaced energy levels
• Electrons fill the lowest-energy bands first
• The highest energy band with electrons in it is called the
valence band
• If the valence band is not full, electrons can move from atom
to atom. This is the case for conductors
• Electrons can not move in filled bands. Thus electrons in
semiconductors must gain energy (usually from thermal
sources) to move from atom to atom.
What else have we learned today?
• In many, ohmic, materials, current is proportional to
voltage:
V = iR
• Resistance is proportional to the length of an object and
inversely proportional to cross-sectional area:
R = rL/A
• The constant of proportionality here is called the
resistivity. It is a function of material and temperature.
• The resistivity of conductors increases with temperature
since atomic vibrations increase.
• The resistivity of semiconductors decreases with
temperature since more conduction electrons exist, and this
effect overshadows the vibrations.
Before the next class, . . .
•
•
•
•
Start (and finish?) Homework 16
Do Activity 14 Evaluation
Read Chapters 3 and start 6 in Turton
Do Reading Quiz
Finish The Activity
Remember,
i = NqAvd, so
R a 1/N, 1/A, 1/q, 1/vd